Questions in Geometric Group Theory — Q 1.10

Definitions and mathematical statement

Let $G$ be a finitely generated word-hyperbolic group. Thus, for a finite symmetric generating set, its Cayley graph with unit edges has $\delta$-thin geodesic triangles for some finite $\delta$. Let $H\leq G$ be a finitely presented subgroup, with its own finite generating set and finite list of defining relations.

Assume that every element of $G$ has a positive power lying in $H$: \[ \forall g\in G\ \exists n(g)\in\mathbb Z_{\geq1} \quad g^{n(g)}\in H. \] Must the index $[G:H]$, the number of left cosets of $H$, be finite?

The exponent is allowed to depend on $g$; there is no common exponent in the hypothesis. The subgroup is not assumed normal. Consequently the assertion cannot in general be rephrased as a torsion statement about a quotient group $G/H$, since that quotient need not be a group. When $H$ is normal, the displayed condition does say that every element of the quotient has finite order, but finite presentation of the subgroup is still a separate hypothesis.

The reverse implication is elementary: if $H$ has finite index, the sequence of cosets $H,gH,g^2H,\ldots$ repeats, forcing a positive power of $g$ into $H$. The question asks for the converse in the stated hyperbolic and finite-presentation setting. It does not merely ask whether $H$ meets every conjugacy class or contains powers of a fixed generating set.

Short English statement

Must a finitely presented subgroup of a hyperbolic group have finite index if it contains a positive power of every group element?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1100110 (AMR-010-0110), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Bestvina - Questions in Geometric Group Theory (pdf) (2004), Question 1.10 (PDF page 3). Original source indexed by AMR-010-0110. https://www.math.utah.edu/~bestvina/eprints/questions-updated.pdf.